A Monte Carlo Simulator for Non-contact

Mode Atomic Force Microscopy

Lado Filipovic1,2 and Siegfried Selberherr1

1 Institute for Microelectronics, Technische Universitat Wien,Gußhausstraße 27–29/E360, A-1040 Wien, Austria

{filipovic,selberherr}@iue.tuwien.ac.at2 Christian Doppler Laboratory for Reliability Issues in Microelectronics at the

Institute for Microelectronics, Wien, Austria

Abstract. Nanolithography using Non-Contact Mode Atomic Force Mi-croscopy (NCM-AFM) is a promising method for the manufacture ofnanometer sized devices. Compact models which suggest nanopatternedoxide dots with Gaussian or Lorentzian profiles are implemented in aMonte Carlo simulator in a level set environment. An alternative tocompact models is explored with a physics based Monte Carlo model,where the AFM tip is treated as a point charge and the silicon waferas an infinite conducting plane. The strength of the generated electricfield creates oxyions which accelerate towards the silicon surface andcause oxide growth and surface deformations. A physics based model ispresented, generating an oxide dot based on the induced surface chargedensity. Comparisons to empirical models suggest that a Lorentzian pro-file is better suited to describe surface deformations when compared tothe Gaussian profile.

1 Introduction

Atomic Force Microscopy (AFM) [1] was developed in 1986 to measure protu-berances and depressions on a nanometer sized section of a desired surface. Forthis purpose AFM has been a useful tool in physics, chemistry, biology, biochem-istry, as well as in the semiconductor industry. This method utilizes the Van derWaals interaction between the tip and a sample surface to determine the surfaceproperties [11]. Several years after the development of AFM, it was noted thatthis method can also be implemented to pattern semiconductor or metal surfaceswith nanometer scale precision for processes such as local anodic oxidation ofsilicon [4]. With the ability to pattern nanoscale devices, the importance of AFMin the semiconductor industry increased significantly. There are various modesin which AFM can operate. The most commonly used for local anodic oxidationof semiconductor surfaces are contact mode (CM-AFM) and non-contact mode(NCM-AFM) [2–4]. NCM-AFM allows narrow patterns to be generated at rela-tively high speed, making it the preferred method for local anodic nanooxidationof silicon surfaces [7].

I. Lirkov, S. Margenov, and J. Wasniewski (Eds.): LSSC 2011, LNCS 7116, pp. 447–454, 2012.c© Springer-Verlag Berlin Heidelberg 2012

448 L. Filipovic and S. Selberherr

Over the past decades the use of NCM-AFM as a tool for the local anodicoxidation of semiconductor surfaces has significantly increased, generating a de-mand for a simulation tool which predicts semiconductor surface deformationdue to AFM processing. Conventional photolithographic methods are unable toaccurately describe the processing steps necessary to generate a nanoscale de-vice using AFM. Having the ability to simulate the AFM process is essentialin order to understand the extent of devices which can be manufactured usingthis method. In this work, we present an analytical model implemented using aMonte Carlo simulator in a level set environment [6]. The level set simulator isan existing silicon process simulator which describes surface deformations afterconventional processing steps are performed on a silicon surface [5]. The final de-formation of a silicon surface after local anodic nanooxidation using NCM-AFMcan either have a Gaussian [10] or a Lorentzian [8] profile.

We further explore an alternative to this compact model by developing aphysics based Monte Carlo model, where the AFM needle tip is treated as apoint charge which generates an electric field towards the silicon surface. Thesilicon surface is treated as an infinite conductive plane, since the silicon waferextends to lengths much longer than the nanoscale AFM tip distance. With thisphysics based Monte Carlo model, the validity of the available analytical modelis explored and the final shape of a silicon wafer dot after applying AFM nano-lithographic processing steps is generated. Using the results, a suggestion will bemade whether to use a Gaussian or Lorentzian profile for the analytical model.The processing steps for a physics based Monte Carlo approach are introduced.

2 Non-Contact Mode Atomic Force Microscopy

AFM is a relatively modern lithographic technique, capable of generating pat-terns on semiconductor surfaces through the application of an electric field. It isa very useful tool for local anodic nanooxidation of silicon surfaces. Fig. 1 showsthe schematic of NCM-AFM, where the tip is placed near the silicon surface anda negative bias voltage, relative to the silicon potential, is applied. An electricfield is generated between the AFM tip and the silicon wafer. The high electricfield near the AFM tip causes the creation of oxyions (O− and OH−) from theair ambient, which are then accelerated towards the silicon surface, away fromthe negatively charged AFM tip. The oxyions react with the silicon substrate,resulting in the generation of silicon dioxide which is simultaneously expandedinto the silicon substrate and the ambient. A detailed explanation of the AFMmethod and its use to physically or chemically modify surfaces at nanometerscales can be found in [11].

3 Empirical Atomic Force Microscopy Model

An empirical model for the generation of an oxide dot using local anodic nano-oxidation of silicon surfaces with the NCM-AFM method was presented in [2].

A Monte Carlo Simulator for NCM-AFM 449

Fig. 1. Basic schematic for the local anodic nano-oxidation of silicon surfaces using NCM-AFM

Fig. 2. AFM tip is treated as apoint charge in order to create aphysics based Monte Carlo simu-lation model

It is suggested that the width and height of an oxide dot have a linear depen-dence on the applied bias voltage, while a logarithmic dependence exists for thepulse duration. An empirical equation which governs the height of the oxide dotproduced with NCM-AFM mode has been presented in [2]

h (t, V ) = h0 (V ) + h1 (V ) ln (t) , (1)

where h0 (V ) = −2.1 + 0.5V − 0.006V 2 and h1 (V ) = 0.1 + 0.03V − 0.0005V 2,and the size, voltage, and time are expressed in nanometers, volts, and seconds,respectively. Similarly, the equation which governs the width of the oxide dot,represented as the full width at half maximum (FWHM) is

w (t, V ) = w0 (V ) + w1 (V ) ln (t) , (2)

where w0 (V ) = 11.6 + 9V and w1 (V ) = 2.7 + 0.9V , and the size, voltage, andtime are expressed in nanometers, volts, and seconds, respectively.

3.1 Monte Carlo Simulation Approach

An analytical simulator for NCM-AFM was created in the level set environmentdescribed in [6] with a Monte Carlo method as shown in Fig. 3. The Monte Carlotechnique is performed by generating a desired number of particles, distributedusing a selected probability distribution along a plane parallel to the siliconsurface, ensuring that the FWHM fits with (2). Each particle is acceleratedtowards the silicon surface. After a collision with the surface, the silicon dioxideis expanded by a height determined by (1) and the total number of particles.After all the particles have been simulated, the result is a surface topographyof a nanodot with a height and width dependent on (1) and (2), respectively.An example from [2] is simulated at a bias voltage of 20V and a pulse time of0.125ms, resulting in a height and width of 1nm and 5.6nm, respectively, shownin Fig. 4.

The results of the experiments from [2] and the analytical model were usedin order to construct a physics based model for NCM-AFM in Section 4.

450 L. Filipovic and S. Selberherr

Fig. 3. Flow chart for the analytical Monte Carlo NCM-AFM simulator

Fig. 4. AFM simulation: bias voltage is 20V and pulse width is 0.125ms. Top surfaceis the oxide-ambient interface, while the bottom surface is the oxide-silicon interface.

3.2 Lorentzian versus Gaussian Profiles

There are a very limited number of simulators which can deal with NCM-AFMnanooxidation, therefore there is not a clear consensus on the exact structureof the final oxide dot. The important feature of the nanodot is its height andFWHM. However, in order to create a simulator for NCM-AFM, more informa-tion is required. Some literature states that the final oxide dot approximatelyfollows a Gaussian curvature [10], while some suggest a Lorentzian profile [8].The differences between the two options are shown in Fig. 5 where the cross-section of a NCM-AFM simulation is depicted. The simulation is performed withthe bias voltage set to 20V and a pulse time of 0.125ms. According to the modeldescribed in Section 3.1 a height of 1nm and a width of 5.6nm are achieved.

From Fig. 5 it is evident that both curves have the same height and width,since the points of intersect for the two curves are at ± 1

2FWHM . The differenceis that the Lorentzian profile suggests that the oxide expands laterally near thesilicon interface, while the Gaussian profile has a steeper curve, resulting inslightly more oxide in the top half of the nanodot.

A Monte Carlo Simulator for NCM-AFM 451

Fig. 5. Cross-section of the oxide pattern when using Gaussian (red-dashed line) dis-tribution and Lorentzian (blue-solid line) distribution

4 Electric Field and Ambient Profiles

In order to simulate the AFM oxidation process with a physical model, the tipof the cantilever from Fig. 1 is treated as a negatively charged stationary dot,while the silicon wafer is treated as an infinitely long conducting plane, shownin Fig. 2. The generated oxyions (particle p from Fig. 2) will move through theambient towards the silicon surface aided by the electric field. It is assumed thatthe silicon surface is on the x-y plane, at z=0, the origin p(0,0,0) is located on thesilicon surface directly below the charged dot, and the charged dot is at a distanced away from the silicon surface. The equation for the strength of the electric fieldat a point p(x,y,z) is well known. It is calculated by assuming the presence ofan oppositely charged stationary dot located at (0,0,-d) and implementing theimage charge method [9]. This method is used in order to calculate the effectivecharge of the AFM needle tip, the electric field between the AFM tip and thesilicon surface, as well as the surface charge density on the silicon surface. Theequation which governs the electric potential at a point p(x,y,z) is

V (−→p ) = k

[Q

(x2+y2+(z−d)2)1/2− Q

(x2+y2+(z+d)2)1/2

], (3)

where k = 1/ (4πε0), Q is the charge at a distance d from the surface. Thevoltage at p(0,0,d) is known, since this is the applied bias voltage. Using (3), wecan find the effective dot charge Q to simulate the AFM tip.

Ex = kQ

[x

(x2+y2+(z−d)2)3/2 − x

(x2+y2+(z+d)2)3/2

]

Ey = kQ

[y

(x2+y2+(z−d)2)3/2 − y

(x2+y2+(z+d)2)3/2

]

Ez = kQ

[z−d

(x2+y2+(z−d)2)3/2 − z+d

(x2+y2+(z+d)2)3/2

](4)

The induced surface charge density is represented as σ (x, y, 0) = ε0Ez (x, y, 0),leading to the expression

σ (x, y, 0) = − dQ

2π (x2 + y2 + d2)3/2. (5)

452 L. Filipovic and S. Selberherr

Fig. 6. Shape of the induced surface charge density above the silicon surface

Using the example from [2], modeled with the empirical simulator from Section 3,(3), and (5) we developed a physics based Monte Carlo model for the final patternof the oxide nanodot.

4.1 Surface Charge Density Modeling

In order to obtain the final pattern for an oxide nanodot generated using NCM-AFM, the physical equations of Section 4 will be implemented for the examplefrom Section 3.1. The initial steps used to generate an oxide nanodot were:

1. Calculate Q based on the applied voltage (20V) and (3) in order to obtaina dot charge of -2.2×10−19C.

2. Using (5) and the known value of the FWHM (5.6nm), calculate the effectivelocation of the dot charge above the silicon surface, d=3.7nm.

3. Using the effective location of the dot charge above the silicon surface, gen-erate an induced surface charge density curve, shown in Fig. 8 and Fig. 9.

4. Use the Monte Carlo rejection technique in order to distribute particlesaround the AFM tip according to the induced surface charge density, asshown in Fig. 6 and further detailed in Section 5.

5 Monte Carlo Model Implementation

The implementation of the Monte Carlo rejection technique in order to distributea desired number of particles according to the electric field strength and inducedsurface charge density is as follows:

1. Generate an evenly distributed particle at position p0(x0,y0,z0), located on aplane parallel to the silicon surface. x0 and y0 are evenly distributed randomvariables, while z0=d is the effective vertical position of the static dot charge.

2. Calculate the induced surface charge density σ (x, y, 0) at p0(x0,y0,0) using(5).

3. Calculate the maximum induced surface charge density σmax (0, 0, 0), at po-sition pmax(0,0,0).

4. Generate an evenly distributed random number ρ between 0 and σmax.5. If ρ > σ (x, y, 0) remove the particle and repeat the procedure from step 1.

If ρ ≤ σ (x, y, 0) continue to the next step.

A Monte Carlo Simulator for NCM-AFM 453

Fig. 7. Surface topography obtained with the presented model

6. Accelerate the particle towards the silicon surface along the vertical direc-tion, until it collides with the surface.

7. At the impact location, advance the ambient-oxide interface towards theambient while advancing the oxide-silicon interface towards the silicon.

8. If the number of particles is 0 the simulation is complete. Otherwise reducethe number of particles by 1 and repeat the procedure from step 1.

The surface topography, following the above simulation steps, is shown in Fig. 7,where the top and bottom surfaces are the final oxide-ambient and oxide-siliconinterfaces, respectively. The induced surface charge density curve is compared toempirical simulations using Gaussian and Lorentzian distributions in Fig. 8 andFig. 9, respectively. It is immediately evident that the Lorentzian distribution isa better fit to the physics based model. The Gaussian distribution has a sharpslope, which fails to properly show the lateral extension of the oxide nanodot.

-5 -4 -3 -2 -1 0 1 2 3 4 5Distance Along Surface (nm)

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

Indu

ced

Surf

ace

Char

ge D

ensi

ty

Charge densityGaussian estimate

Fig. 8. Comparison to simulation with aGaussian distribution

-5 -4 -3 -2 -1 0 1 2 3 4 5Distance Along Surface (nm)

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

Indu

ced

Surf

ace

Char

ge D

ensi

ty

Charge densityLorentzian estimate

Fig. 9. Comparison to simulation with aLorentzian distribution

454 L. Filipovic and S. Selberherr

6 Conclusion

Modern lithographic methods are reaching the edge of their potential and nano-lithography using NCM-AFM is a promising alternative for the manufacture ofnanometer sized devices. A compact Monte Carlo simulator for NCM-AFM is im-plemented to simulate nanodots based on empirical models. Various publicationssuggest that surface deformations due to AFM have a Gaussian or Lorentzianprofile. A physics based Monte Carlo model, treating the AFM tip as a staticcharge which generates an electric field towards the silicon surface, was created.Using a Monte Carlo rejection technique, the induced surface charge density isused as the basis of the final topography. The model shows a better fit to theLorentzian distributed empirical model when compared to the Gaussian model.

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